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[pilot]

How do you convert MACH to true airspeed and vice versa?

[admin]

In order to convert the true airspeed to a Mach number, which is the percent of speed of sound, we have to find what the speed of sound is for the current conditions.
The speed of sound depends on the ambient temperature (and not directly on altitude):

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Speed of sound in knots = 39 X SQRT of outside air temperature in Kelvin

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temp in Kelvin = Temp in Celsius + 273

For example, if the outside air temperature is -50 deg C and TAS is 400 knots:
Speed of sound = 39 X SQRT(-50+237) = 533 knots
Since our true airspeed is 400 we are cruising at 400/533 = 0.75 Mach which is 75% the speed of sound for this temperature.

You can see from here how easy it is to calculate this problem in reverse.
Our Mach number is .75 OAT is -50 deg. C.
Speed of sound at -50 deg C = 39 X SQRT(-50+237) = 533 knots
0.75 Mach X 533 = 400 knots TAS